A192948 -id:A192948 - cp's OEIS frontend

A192945 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^2 - 1)^n.

1, 1, 2, 9, 50, 311, 2072, 14460, 104346, 772255, 5829538, 44710705, 347424376, 2729299748, 21640457360, 172957598120, 1391926695402, 11270059892943, 91740990170150, 750364940281275, 6163650579487170, 50824871829196575

Offset: 0

Paul D. Hanna, Jul 13 2011

nonn

Compare to a g.f. of the Catalan numbers: C(x) = 1 + x*Sum_{n>=0} (C(x) - 1)^n.

			G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 50*x^4 + 311*x^5 + 2072*x^6 + ...
where (A(x) - 1)*(2 - A(x)^2) = x
and A(x - 2*x^2 - x^3) = 1 + x.
Related expansions:
(A(x)^2-1) = 2*x + 5*x^2 + 22*x^3 + 122*x^4 + 758*x^5 + 5047*x^6 + ...
(A(x)^2-1)^2 = 4*x^2 + 20*x^3 + 113*x^4 + 708*x^5 + 4736*x^6 + ...
(A(x)^2-1)^3 = 8*x^3 + 60*x^4 + 414*x^5 + 2909*x^6 + 20970*x^7 + ...
(A(x)^2-1)^4 = 16*x^4 + 160*x^5 + 1304*x^6 + 10184*x^7 + ...
Also,
A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 122*x^4 + 758*x^5 + 5047*x^6 + ...
A(x)^3 = 1 + 3*x + 9*x^2 + 40*x^3 + 222*x^4 + 1380*x^5 + 9191*x^6 + ...
where 2 + x = 2*A(x) + A(x)^2 - A(x)^3.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The three 'R's and the Riordan dual, arXiv:1609.01193 [math.CO], 2016.

Cf. A192946, A192947, A192948.

Flatten[{1,Table[1/n*Sum[Binomial[n+k-1,n-1]*Sum[Binomial[i,n-k-i-1]*Binomial[n+k+i-1,n+k-1],{i,Floor[(n-k-1)/2], n-k-1}],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec after Vladimir Kruchinin, Nov 20 2012 *)

a(n):=if n=0 then 1 else 1/n*sum(binomial(n+k-1,n-1) *sum(binomial(i,n-k-i-1)*binomial(n+k+i-1,n+k-1),i,ceiling((n-k-1)/2), n-k-1),k,0,n-1); /* Vladimir Kruchinin, Oct 11 2011 */

{a(n)=local(A=1+x);for(i=1,n,A=1+x*sum(m=0,n,(A^2-1+x*O(x^n))^m));polcoeff(A,n)}

{a(n)=local(A=1+serreverse(2*x-x*(1+x)^2+x^2*O(x^n)));polcoeff(A,n)}

G.f. A(x) equals the formal inverse of function (x-1)*(2-x^2).
G.f. satisfies: A(x) = 1 + x/(2 - A(x)^2).
G.f.: A(x) = 1 + Series_Reversion( 2*x - x*(1+x)^2 ).
a(n) = (1/n)*Sum_{k=0..n-1} binomial(n+k-1, n-1)*Sum_{i=ceiling((n-k-1)/2)..n-k-1} binomial(i, n-k-i-1)*binomial(n+k+i-1, n+k-1), n > 0, a(0)=1. - Vladimir Kruchinin, Oct 11 2011
Recurrence: 8*(n-1)*n*a(n) = 34*(n-1)*(2*n-3)*a(n-1) + 3*(3*n-7)*(3*n-5)*a(n-2). - Vaclav Kotesovec, Nov 20 2012
a(n) ~ 1/3*sqrt(7/6 - 17/(6*sqrt(7)))*((17 + 7*sqrt(7))/4)^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 20 2012

A192946 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^3 - 1)^n.

Original entry on oeis.org

1, 1, 3, 21, 181, 1746, 18039, 195214, 2184381, 25067856, 293420578, 3489516381, 42044519283, 512146618088, 6296546349018, 78031090301868, 973723814391957, 12224652295383324, 154299365902579044, 1956876044969421604, 24924046596321581940

Offset: 0

Paul D. Hanna, Jul 13 2011

nonn

Compare to a g.f. of the Catalan numbers: C(x) = 1 + x*Sum_{n>=0} (C(x) - 1)^n.

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 181*x^4 + 1746*x^5 + 18039*x^6 +...
where (A(x) - 1)*(2 - A(x)^3)  = x
and A(x - 3*x^2 - 3*x^3 - x^4) = 1+x.
Related expansions.
(A(x)^3-1) = 3*x + 12*x^2 + 82*x^3 + 705*x^4 + 6792*x^5 + 70122*x^6 +...
(A(x)^3-1)^2 = 9*x^2 + 72*x^3 + 636*x^4 + 6198*x^5 + 64396*x^6 +...
(A(x)^3-1)^3 = 27*x^3 + 324*x^4 + 3510*x^5 + 38475*x^6 +...
(A(x)^3-1)^4 = 81*x^4 + 1296*x^5 + 16632*x^6 + 203148*x^7 +...
Also,
A(x)^3 = 1 + 3*x + 12*x^2 + 82*x^3 + 705*x^4 + 6792*x^5 + 70122*x^6 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 124*x^3 + 1067*x^4 + 10284*x^5 + 106200*x^6 +...
where 2+x = 2*A(x) + A(x)^3 - A(x)^4.

G. C. Greubel, Table of n, a(n) for n = 0..875
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A192945, A192947, A192948.

CoefficientList[1+InverseSeries[Series[2*x-x*(1+x)^3,{x,0,20}],x],x] (* Vaclav Kotesovec, Sep 17 2013 *)

{a(n)=local(A=1+x);for(i=1,n,A=1+x*sum(m=0,n,(A^3-1+x*O(x^n))^m));polcoeff(A,n)}

{a(n)=local(A=1+serreverse(2*x-x*(1+x)^3+x^2*O(x^n)));polcoeff(A,n)}

G.f. A(x) equals the formal inverse of function (x-1)*(2-x^3).
G.f. satisfies: A(x) = 1 + x/(2 - A(x)^3).
G.f.: A(x) = 1 + Series_Reversion( 2*x - x*(1+x)^3 ).
Recurrence: 108*(n-2)*(n-1)*n*(9*n-25)*a(n) = 18*(n-2)*(n-1)*(702*n^2 - 3003*n + 2924)*a(n-1) + 3*(n-2)*(3375*n^3 - 22875*n^2 + 50704*n - 36656)*a(n-2) + 8*(2*n-5)*(4*n-13)*(4*n-11)*(9*n-16)*a(n-3). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ 1/48*sqrt(6)*sqrt(1271 - (2661+32096*sqrt(2))^(2/3) + 3*(2661+32096*sqrt(2))^(1/3))/((2661 + 32096*sqrt(2))^(1/6)) * ((3*(841+34*sqrt(2))^(2/3) + 267 + 26*(841+34*sqrt(2))^(1/3)) / (841+34*sqrt(2))^(1/3)/6)^n / (n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Sep 17 2013

A192947 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^4 - 1)^n.

Original entry on oeis.org

1, 1, 4, 38, 444, 5805, 81284, 1192144, 18078660, 281172017, 4460264072, 71886775636, 1173832034804, 19377733213699, 322866234066016, 5422493523853024, 91701823351874276, 1560232214582865621, 26688686144512908492

Offset: 0

Paul D. Hanna, Jul 13 2011

nonn

Compare to a g.f. of the Catalan numbers: C(x) = 1 + x*Sum_{n>=0} (C(x) - 1)^n.

			G.f.: A(x) = 1 + x + 4*x^2 + 38*x^3 + 444*x^4 + 5805*x^5 +...
where (A(x) - 1)*(2 - A(x)^4) = x
and A(x - 4*x^2 - 6*x^3 - 4*x^4 - x^5) = 1 + x.
Related expansions.
(A(x)^4-1) = 4*x + 22*x^2 + 204*x^3 + 2377*x^4 + 31036*x^5 +...
(A(x)^4-1)^2 = 16*x^2 + 176*x^3 + 2116*x^4 + 27992*x^5 +...
(A(x)^4-1)^3 = 64*x^3 + 1056*x^4 + 15600*x^5 + 232456*x^6 +...
(A(x)^4-1)^4 = 256*x^4 + 5632*x^5 + 98688*x^6 + 1640576*x^7 +...

G. C. Greubel, Table of n, a(n) for n = 0..785

Cf. A192945, A192946, A192948.

CoefficientList[1+InverseSeries[Series[2*x-x*(1+x)^4,{x,0,20}],x],x] (* Vaclav Kotesovec, Sep 17 2013 *)

{a(n)=local(A=1+x);for(i=1,n,A=1+x*sum(m=0,n,(A^4-1+x*O(x^n))^m));polcoeff(A,n)}

{a(n)=local(A=1+serreverse(2*x-x*(1+x)^4+x^2*O(x^n)));polcoeff(A,n)}

G.f. A(x) equals the formal inverse of function (x-1)*(2-x^4).
G.f. satisfies: A(x) = 1 + x/(2 - A(x)^4).
G.f.: A(x) = 1 + Series_Reversion( 2*x - x*(1+x)^4 ).
Recurrence: 2048*(n-3)*(n-2)*(n-1)*n*(150965*n^3 - 1545075*n^2 + 5236171*n - 5868843)*a(n) = 256*(n-3)*(n-2)*(n-1)*(21135100*n^4 - 248013150*n^3 + 1057496990*n^2 - 1920941631*n + 1231831665)*a(n-1) + 32*(n-3)*(n-2)*(204104680*n^5 - 2905360120*n^4 + 16233323207*n^3 - 44427578447*n^2 + 59458979970*n - 31076616060)*a(n-2) + 8*(n-3)*(372581620*n^6 - 6607607250*n^5 + 48312584548*n^4 - 186256666080*n^3 + 398929297639*n^2 - 449551482162*n + 207932551470)*a(n-3) + 5*(5*n - 21)*(5*n - 19)*(5*n - 18)*(5*n - 17)*(150965*n^3 - 1092180*n^2 + 2598916*n - 2026782)*a(n-4). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 35/8 + 1/(8*sqrt(3/(4351 + (6495403 + 452895*sqrt(831))^(1/3)/2^(2/3) - 40027/(12990806 + 905790*sqrt(831))^(1/3)))) + 1/2*sqrt(4351/24 - (6495403 + 452895*sqrt(831))^(1/3)/(48*2^(2/3)) + 40027/(48*(12990806 + 905790*sqrt(831))^(1/3)) + 27661/4*sqrt(3/(4351 + (6495403 + 452895*sqrt(831))^(1/3)/2^(2/3) - 40027/(12990806 + 905790*sqrt(831))^(1/3)))) = 18.6600216048327281... is the root of the equation -3125 - 19744*d - 43264*d^2 - 35840*d^3 + 2048*d^4 = 0 and c = 0.047032341973499367520535028629... - Vaclav Kotesovec, Sep 17 2013

A232192 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^n - 1)^n.

Original entry on oeis.org

1, 1, 1, 5, 44, 519, 7590, 132347, 2689046, 62644234, 1651650774, 48731341965, 1592908456996, 57173688136781, 2235773294509565, 94608603077007214, 4306708055122614542, 209823573154587335730, 10892496561736261641371, 600171728539156939466278

Offset: 0

Paul D. Hanna, Nov 20 2013

nonn

			G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 44*x^4 + 519*x^5 + 7590*x^6 + 132347*x^7 + 2689046*x^8 + 62644234*x^9 + 1651650774*x^10 +...
where
A(x) = 1 + x + x*(A(x)-1) + x*(A(x)^2-1)^2 + x*(A(x)^3-1)^3 + x*(A(x)^4-1)^4 + x*(A(x)^5-1)^5 + x*(A(x)^6-1)^6 + x*(A(x)^7-1)^7 +...
Also,
A(x) = 1 + x/2  +  x*A(x)/(1 + A(x))^2  +  x*A(x)^4/(1 + A(x)^2)^3  +  x*A(x)^9/(1 + A(x)^3)^4  +  x*A(x)^16/(1 + A(x)^4)^5  +  x*A(x)^25/(1 + A(x)^5)^6  + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..175

Cf. A122400, A192945, A192946, A192947, A192948.

{a(n)=local(A=1+x); for(i=1, n, A=1+x*sum(m=0, n, (A^m-1+x*O(x^n))^m)); polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

G.f. satisfies:
(1) A(x) = 1 + x*Sum_{n>=0} A(x)^(n^2) / (1 + A(x)^n)^(n+1). - Paul D. Hanna, Mar 31 2018
(2) A(x) = 1 + Series_Reversion(x/G(x))
(3) A(x) = 1 + x*G(A(x)-1)
where G(x) is the g.f. of A122400, the number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.
a(n) ~ c * d^n * n! / n^(3/2), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.12140554666... . - Vaclav Kotesovec, May 07 2014

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A192945 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^2 - 1)^n.

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A192946 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^3 - 1)^n.

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A192947 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^4 - 1)^n.

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A232192 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^n - 1)^n.

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